Exponentiating Higgs

We consider two related formulations for mass generation in the $U(1)$ Higgs-Kibble model and in the Standard Model (SM). In the first model there are no scalar self-interactions and, in the case of the SM, the formulation is related to the normal subgroup of $G=SU(3)\times SU(2)\times U(1)$, generated by $(e^{2\pi i/3}I,-I,e^{\pi i/3})\in G$, that acts trivially on all the fields of the SM. The key step of our construction is to relax the non-negative definiteness condition for the Higgs field due to the polar decomposition. This solves several stringent problems, that we will shortly review, both in the perturbative and non-perturbative formulations. We will show that the usual polar decomposition of the complex scalar doublet $\Phi$ should be done with $U\in SU(2)/Z_2\simeq SO(3)$, where $Z_2$ is the group generated by $-I$, and with the Higgs field $\phi\in R$ rather than $\phi\in R_{\geq0}$. As a byproduct, the investigation shows how Elitzur theorem may be avoided in the usual formulation of the SM. It follows that the simplest lagrangian density for the Higgs mechanism has the standard kinetic term in addition to the mass term, with the right sign, and to a linear term in $\phi$. The other model concerns the scalar theories with normal ordered exponential interactions. The remarkable property of these theories is that for $D>2$ the purely scalar sector corresponds to a free theory.


Introduction
The Higgs mechanism [1]- [6] is a basic step in the formulation of the Standard Model [7,8]. This has been confirmed by the spectacular experimental results at LHC [9,10]. Despite this, there are still some open questions. The most important one is that the vev of the Higgs field is evaluated at the classical level. On the other other hand, there are models with non-trivial minima for the classical potential, with no order parameter. The point is that spontaneously symmetry breaking is a strictly nonperturbative phenomenon, concerning infinitely many degrees of freedom. As such, even radiative corrections to φ should be considered with particular attention. In this respect, one should also recall that, against the evidence coming from the perturbative expansion, λφ 4 is believed to be a free theory. A related aspect is that the mass term of the initial lagrangian has the opposite sign. Other questions concern the hierarchy problem, with the LHC data that do not seem to confirm that supersymmetry may be the answer, and understanding whether there is a mechanism giving mass to the Higgs. In trying to answer the above questions one should keep in mind that the main reason for the use of Higgs model is the one of providing a nontrivial vev for the scalar field. The natural question is then if there is a simpler way to get such a vev that may avoid the above mentioned questions. This is of considerable experimental interest, since a possible alternative to the standard Higgs mechanism can be verified in future experiments at LHC. In particular, it should be possible to make the crucial check of the η 3 term, and, even if much more difficult, of the η 4 self-interaction. We will start the investigation by considering the parametrization of the Higgs field Φ. In particular, we will consider a simple model, which is naturally suggested once one chooses the parametrization .
The point is that, as a matter of fact, φ is tacitly considered to take real values rather than non-negative ones. Such a choice is usually justified by the fact that one is considering perturbation theory around a minimum. In the next section we will consider a more direct justification for considering φ a real field. This suggests a simple form for the scalar lagrangian of the Standard Model. Next, we will consider exponential interactions. In this respect, in [11], it has been shown that these lead to a nontrivial φ . An outcome of [11], is that there appear terms coming from the normal ordering of the exponentiated scalar.
In this respect, we note that the interaction term V in the path-integral formulation can be seen as the vev, in the free theory in an external source, of exp( d D x : V (φ) :), that is It is worth noticing that, as emphasized in [12], the difficulties in quantizing some nonrenormalizable field theories, concern the non-uniqueness of the solution, rather than its existence. In such a context, let us remind that in [13], using the ultraviolet cutoff γ −N , γ > 1, N > 0, have been investigated scalar theories with interaction λ : exp(αφ) :. It turns out that for d > 2, for all α, and for d = 2, with |α| > α 0 , the Schwinger functions converge to the free Schwinger functions. The essential point in the investigation of [13] is that ∆ F,Λ (0), with ∆ F,Λ (x) the Feynman propagator with cut-off on the momenta, grows sufficiently fast to kill the fluctuations of φ, so that : exp(αφ) := exp(− α 2 2 ∆ F,Λ (0)) exp(αφ) vanishes in the limit Λ → ∞. We are not aware if such findings have been reproduced in the standard lattice regularization. However, as we will see, we will get nontrivial expectation values of φ. It would be of considerable interest if this may correspond to a gaussian theory. We will consider the D-dimensional euclidean partition function of the scalar theory with potential V (φ) = e −µ D d D x exp(−αφ(x)) , and then consider This can be seen as filling-in the free vacuum in an external source by scalar modes. By using an iterative algorithm, reproducing the Wick theorem, we derive a simple way to implement such a normal ordering in the path-integral approach. The results in [11] straightforwardly imply that is the Feynman propagator. It turns out that Z R [J] generates the lowest order contributions in α to the N-point point function. In particular Next, we consider the lagrangian density and then, as usual, parameterize the scalar doublet in polar coordinates. In the unitary gauge, the scalar part of L Φ reduces to We will see that this leads to generates only the lowest order contribution to the N = 2 point function We then consider the model as an effective theory. An example of possible choice is to identify ν with ∆ F,Λ (0)/m, so that

A simple model
In the unitary gauge, the Higgs field Φ has only one non-vanishing component, the field the field η is usually considered as taking real values. This seems a subtle point since the lagrangian of the Standard Model contains the term η 3 and the linear one in η in the Yukawa couplings. The usual argument is that this is justified when considering perturbation around a minimum. This would suggest considering the lagrangian density In the unitary gauge, the purely scalar part reduces to The corresponding vev for φ is v = 2m, so that, if perturbation theory justifies taking φ real, one would get the free lagrangian for η = φ − v. It is then clear that it would be desirable finding a more natural way to justify the parametrization with φ ∈ R, so that one may consider (2.4) with |φ| replaced by φ. In this respect, note that resembles the decomposition of complex numbers z = χe iθ , with χ ∈ R and θ ∈ [π/2, π/2), a choice that avoids the multivaluedness of the arctangent parametrization of θ. As in the standard polar decomposition, one may then extend θ to R. Now the transformation z → −z can be obtained either by χ → −χ or by θ → θ+(2k +1)π, k ∈ Z. This represents an alternative to the periodicity θ → θ + 2kπ, k ∈ Z of the polar decomposition. Since φ in (2.2) is gauge invariant, any V (φ) is a gauge invariant potential. With the choice (2.2) one has to remind that φ = e kπi |φ| = e kπi φ 2 , k ∈ Z, so that φ = (φ 2 ) We stress that a gauge transformation of Φ(x) by a sign corresponds to U(x) → e 2(k+1)πi U(x), so that U(x) † U(x) = e 0 , and, as obvious, (Φ † Φ) 1 2 is invariant. In the following we use the parametrization (2.2). Let us consider the lagrangian density for the scalar field Φ and note that 2) it follows that, in the unitary gauge, the lagrangian density describing the purely scalar sector is Note that, since the theory is free, it follows that φ coincides with the value of φ that minimizes Let us recall that it is a general fact that higher order derivatives of the logarithm of the partition function, in an external source J, evaluated at J = 0, coincide with the connected correlators of η = φ − φ . For example, This identifies η as the true scalar field, and the lagrangian density becomes the free one 3 Exponential interactions Let us introduce some notation, which follows the one in Ramond's book [14], and shortly reviewing the investigation in [11]. Consider the partition function in the D-dimensional euclidean space and denote by f (x 1 , . . . , x n ) integration of f over x 1 , . . . , x n . Let be the Feynman propagator and set The starting point in [11] has been the observation that Schwinger's trick can be extended to get exact results. In particular, it has been observed that exponential interactions can be obtained by acting on exp(−Z 0 [J]) with the translation operator. Consider the potential investigated in [11] with the opposite sign of α The corresponding partition function (we drop the constant N) is Then, we use [11] exp Therefore, The terms exp kα 2 2 ∆ F (0) and exp(α 2 k j>l ∆ F (z j − z l ) are related to normal ordering. In this respect, note that (3.7) corresponds to the expansion 12) where the vacua are the ones of the free scalar theory coupled to the external source J. The fact that the normal ordering problem is the cause of some of the infinities arising in perturbation theory, suggests considering To check the consequence of taking the normal ordering, note that Therefore, It follows that the expansion on right hand side in (3.13) exponentiates. Actually, (3.10), (3.12), (3.13) and (3.16) yield Interestingly, removing the term exp(α 2 k j>l ∆ F (x j − x l )), coming from the normal ordering in (3.15), is equivalent to remove a term exp(−α δ δJ ) in (3.7). To show this, recall that for any suitable function F , if A and B are operators, then A −1 F (B)A = F (A −1 BA). Therefore, where the dots denote the terms in (3.10) coming from the expansion and note that by (3.18) that satisfies the equation of motion (3.24) it follows that at the first order in α Such a relation follows, for example, by that is and by (3.18), for N > 1, Note that higher order contributions in α come from the expansion (3.22).

v = 2m
The above model can be extended to more general interactions, such as In order to find the explicit expression of W R [J] in the case of the potential (4.1), one first notes that then, uses (3.19) iteratively. In the first step one has Repeating this for the remaining n − 1 terms in (4.2), makes it clear that is obtained from W [J] by removing, from the final expression, the term exp n k=1 α k δ δJ on the right hand side, and by canceling the exp n k=1 α 2 k ∆ F (0) term. Such a cancelation is equivalent to relabel each µ k0 by µ k . It follows that We note that taking the normal ordering of exp( d D xV (φ)) may lead to well-defined Z R [J], even in cases when V (φ) is unbounded below. A particularly interesting case is the four-dimensional potential where ν is a dimensionless scale. By (4.5), we have Repeating the analysis leading to (3.27), at the zero order in ν −1 , (4.7) yields 8) so that, at the same order, in agreement with the LHC data. Remarkably, we have lim ν→∞ V (φ) = −2m 3 φ , (4.10) so that, in this limit, (4.8) corresponds to the value of φ that minimizes m 2 φ 2 /2 + V (φ). The resulting lagrangian density coincides with (2.4).
Considering the model as an effective theory suggests relating ν with the momentum cutoff Λ. To select a natural choice, note that since φ(x)φ(x) is the amplitude of finding a particle at x at the time t which is in the same point at the same time, it follows that in a finite theory G(x, x) = χ(x)χ(x) should be one. This implies that the correct normalization is G(x, x) = 1 . Note that for large Λ the model is well-described by Z R [J].